group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A cohomology theory $E$ is called multiplicative if $E^\bullet(X)$ is not just a graded abelian group, but actually a graded ring.
This is in particular the case when the spectrum $E$ representing the cohomology theory is an E-∞ ring spectrum.
For multiplicative cohomology theories one can consider
See also